Jefferson and Westchester are a long way off from each other, possible, but would think buried as Barnard. There's a Hannah Hunt that died in Cortlandt, Westchester County, NY that matches to death in 1862 not 1882 per Probate Records. I can't help but wonder if the FindAGrave is off on date for the Hannah Hunt memorial since there is no photo to verify it (wouldn't be the first time). No mention of name Barnard in those records, they mention a sister Charlotte Sutton, nephew Ellis Travis, others.
Hannah's in many trees with no sources. Some list Josiah vs. John as father. Doesn't look like same person you're seeking. That will in probate on Ancestry if you can view here might at least help you rule out.
The Edward Barnard born in Deerfield, MA in 1788 seems to be attached to the same census records and FindAGrave listing. Father Ebenezer Barnard and Mother Abigail Caitlin (in Massachusetts Town & Vital Records).
Are you sure he was born in Vermont?